HMMT 二月 2019 · 几何 · 第 3 题

HMMT February 2019 — Geometry — Problem 3

Let AB be a line segment with length 2, and S be the set of points P on the plane such that there exists point X on segment AB with AX = 2 P X . Find the area of S .

解析

Let AB be a line segment with length 2, and S be the set of points P on the plane such that there exists point X on segment AB with AX = 2 P X . Find the area of S . Proposed by: Yuan Yao √ 2 π Answer: 3 + 3 Observe that for any X on segment AB , the locus of all points P such that AX = 2 P X is a circle 1 centered at X with radius AX . Note that the point P on this circle where P A forms the largest angle 2 ◦ with AB is where P A is tangent to the circle at P , such that ∠ P AB = arcsin(1 / 2) = 30 . Therefore, ′ if we let Q and Q be the tangent points of the tangents from A to the circle centered at B (call it 1 ′ ω ) with radius AB , we have that S comprises the two 30-60-90 triangles AQB and AQ B , each with 2 √ 1 ◦ ′ 2 area 3 and the 240 sector of ω bounded by BQ and BQ with area π . Therefore the total area 2 3 √ 2 π is 3 + . 3